Pip -rS, 71 bytes

YRPg^sw:C`"\w+"`Wy~w.';.wY$`R$1$2.:$2.$'RVSN:{[_@1MS@_QaFIya]}MUQ@Z V:y

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I'm gonna golf this some more before writing a thorough explanation. The basic idea:

• Repr the input so it's one big string with the names are in quotes but the numbers not
• Loop while there's a pair of names; replace every instance of the first name with the second name in everything before the pair
• Eval to turn the string back into a list again
• Do a bunch of mapping and filtering to extract the scores for each name; sort and print

Pip -rS, 71 53 bytes

{IaNI@ZllPB[a0](lY@Zl@?a1)+:bI+bNEb(ly0):b}MUg^sRVSNl

Try it online!

I'm gonna golf this some more before writing a thorough explanation. The basic idea:

• Keep a list of [name score] pairs in l
• Loop through the lines of input, split on spaces, using MU (for each line, the first item is a and the second item is b):
  • If a is not one of the names in l, append [a 0] to l
  • Add b to the score in l associated with the name a (if b is a name, this adds 0*)
  • If b is not a number, change the name a in l to b
• Once the loop is done, sort l in descending numerical order (since all the names are 0 in numeric context*, this sorts by the scores)

^{* Assuming that no names start with a digit, which I argue is the best Pip-specific interpretation of "whatever the input is, it won't be able to be parsed as a number."}